Asymptotics of Multivariate Sequences: I. Smooth Points of the Singular Variety

Given a multivariate generating function F(z1, …, zd)=∑ ar1, …, rdzr11…zrdd, we determine asymptotics for the coefficients. Our approach is to use Cauchyʹs integral formula near singular points of F, resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers treat singular points of F where the local geometry is more complicated, and for which other methods of analysis are not known.